Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573549 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,11)(9,12)(13,14)(15,22)(16,21)(17,19)(18,25)(20,23)(24,28)(26,27);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(14,19)(20,27)(21,25)(22,28);(7,12)(9,10)(13,15,17,24)(14,20,19,27)(16,23,18,26)(21,28,25,22);(5,7)(6,10)(8,12)(9,11)(14,19)(20,27)(21,25)(22,28);(7,12)(9,10)(13,16,17,18)(14,21,19,25)(15,23,24,26)(20,28,27,22)Polycyclic presentation:
2^9|b^2=fhi,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=eh83 conjugacy classes — view the full character table.
| maximal subgroup of | #72283111, #72283120, #72284363, … |
| maximal normal subgroup of | #72283111, #72283120, #72284363, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6504045);